## Relative Signal Strengths Using Lunar Retroreflectors

### Introduction

These days, all too often, one runs across people who do not believe the Moon landings occurred. There were many Hams who received signals not only from the Apollo spacecraft, but from the ALSEP packages before they were shut down in the late 70's. They could face down most doubters with their compelling evidence. Sadly, these Hams are getting old, with more and more becoming SK's. So it gets ever harder to convince the doubters.

Some doubters are real conspiracy nuts and no amount of argument or evidence will convince them. They are a lost cause. However, there is one piece of evidence still in existence that can convince the more reasonable doubter, namely the retroreflectors. They are used even today by the McDonald Observatory in Texas, for example.

Of course, the first thing doubters do is question whether a retroreflector is necessary to receive a return signal, thus casting its existence into doubt. Fortunately, the numbers show how such a reflector is mandatory for the system to work as it does.

This page is an attempt to show the calculations and numbers resulting from the specifications and measurements involved in this laser range finding activity.

If you see an error or inconsistency, please email me (cfb 002 at austin dot rr dot com). Form the address correctly for email (remove the spaces and substitute '.' for dot, etc.). Apologies for this. It is so munged to foil the damned email spammers.

### Specifications of Actual Equipment Used and Other Real Measurements

• Albedo of Moon Surface = 0.073
• Distance to Moon = 3.6x10^8 Meters

• Laser Pulse Energy = 1500x10^-3 Joules
• Laser Pulse Duration = 200x10^-12 Seconds
• Laser Pulse Rate = 10 Hz
• Laser Pulse Power = 7.5x10^9 Watts
• Average Laser Power Output = 15 Watts

• Diameter of Laser Spot on Moon = 7 Kilometers
• Diameter of Retroreflector Spot on Earth = 20 Kilometers

• Collecting Area of Telescope = 0.37 Square Meters (ie, a telescope with ~0.75 meter diameter mirror obstructed by a 15cm secondary)
• Telescope Optical Path Efficiency = 90%

• Collecting Area of Retroreflector = 0.42 Square Meters
• Retroreflector Efficiency = 90%

### Calculations

These calculations do not take into account a host of variables, not the least of which are atmospheric losses and inability to distinguish random or noise photons from returning laser photons. The use of a "censoring" shutter on the scope, in an attempt to minimize noise photon counting, also causes many returning laser photons to be lost. These measures (among others) can have a large effect on the number of photons counted, so the results below should be considered ballpark figures and used mostly as a relative measure of the return signal strengths rather than as absolute measurements.

#### Without Retroreflector

First, we'll look at the case of the laser illuminating a spot on the Moon without a retroreflector. The light reflecting from the illuminated spot loses coherence and collimation. A spot 7KM in diameter on the Moon is small when viewed from the Earth. Thus, although somewhat simplistic, we can treat it as point source of light shining equally in all directions possible from the Moon's surface (ie, a half sphere of illumination).

1. Given the efficiency of the scope, the power of the beam output by it is:
15 Watts * 0.9 = 13.5 Watts

2. Given the Albedo of the Moon, the average power of the reflected light is:
13.5 * 0.073 = 0.9 Watts

3. The area of the shell of illumination reaching the Earth's distance is:
2 * Pi * (3.6x10^8)^2 ~= 8.1x10^17 Square Meters (a half sphere)

4. The collecting area of the receiving telescope is 0.34 Square Meters. Thus, the ratio of collected light to total light is:
0.34 / 8.1x10^17 ~= 4.2x10^-19 [from 3)]

5. The average power of the energy entering the scope is:
0.9 * 4.2x10^-19 ~= 3.8x10^-19 Watts [from 2. and 4.]

6. Thus, given the scope optics efficiency, the total power collected is:
0.9 * 3.8x10^-19 ~= 3.4x10-19 Watts [from 5]

#### With Retroreflector

Now we'll look at the case where the laser illuminates a Lunar retroreflector. For simplicity's sake, we'll ignore the energy returned from the Lunar surface proper and focus just on the energy returned by the retroreflector.

7. The area of the illuminated spot on the Moon in square meters is:
Pi * 3500^2 ~= 3.8x10^7 Square Meters

8. The ratio of the retroreflector area to total illuminated spot area is:
0.42 / 3.8x10^7 ~= 1.1x10^-8 [from 7.]

9. The power of the laser light collected by the retroreflector is:
13.5 Watts * 1.1x10^-8 = 1.5x10^-7 Watts [from 1 and 8.]

10. The power returned by the 90% efficient reflector is:
1.5x10^-7 * 0.9 ~= 1.3x10^-7 Watts [from 9.]

11. The area of the retroreflected spot on the Earth is:
Pi * 10000^2 ~= 3.1x10^8 Square Meters

12. Again, the collecting area of the receiving telescope is 0.34 Square Meters. Thus, the ratio of collected retroreflected light to total light is:
0.34 / 3.1x10^8 ~= 1.1x10^-9 [from 11.]

13. The average power of the energy entering the scope is:
1.3x10^-7 * 1.1x10^-9 ~= 1.4x10^-16 Watts [from 10. and 12.]

14. Thus, given the scope's efficiency, the average power collected when using a retroreflector is:
0.9 * 1.4x10^-16 Watts = 1.3x10-16 Watts [from 13]

15. Using 10 * log (Pwr1 / Pwr2), the power dB difference is:
10 * log ( 1.3x10^-16 / 3.4x10^-19 ) ~= 26dB [from 6. and 14.]

### Summary Conclusion

The retroreflector produces over two orders of magnitude more return. The 26dB gain is a very large difference in signal strength, especially when operating at the very edges of possible sensitivity. Given how difficult it is to detect the laser's reflected signal when using the retroreflector (0.00013 picoWatts or 130 attoWatts!), it would be unimaginably more difficult without it ( 0.00000034 picoWatts or 0.34 attoWatts!).

In all the above calculations, remember that it is the relative difference between the retroreflector and non-retroreflector cases being emphasized here. The absolute measurements of the return signals are not that important as I have not included the many environmental factors that can affect the values. But in calculating the relative difference, those factors are largely eliminated as they apply to both cases. The relative GAIN is the important number.

One final note: it doesn't matter how strong or weak the returning signal is, it would not be possible to calculate the Moon's distance to centimeter accuracy were it not for the retroreflectors. I have posed this as a problem to a pathological doubter. Thus I shall not yet explain why. For those who want a hint, examine the problems that bedevil the Hams who engage in Moonbounce.